Editing Pythagorean tuning (section)

==History and usage==
The system dates to Ancient Mesopotamia,{{sfn|Dumbrill|1998|p=18}} and consisted of alternating ascending fifths and descending fourths; see {{slink|Music of Mesopotamia|Music theory}}. Within Ancient Greek music, the system had been mainly attributed to [[Pythagoras]] (who lived around 500 BCE) by modern authors of music theory; Ancient Greeks borrowed much of their music theory from Mesopotamia, including the diatonic scale, Pythagorean tuning, and modes. The Chinese [[Shí-èr-lǜ|Shí-èr-lǜ scale]] uses the same intervals as the Pythagorean scale and was invented between 600 BCE and 240 CE.<ref name="B&S" /><ref name="Needham">Needham, Joseph (1962/2004). ''Science and Civilization in China, Vol. IV: Physics and Physical Technology'', pp.&nbsp;170–171. {{ISBN|978-0-521-05802-5}}.</ref>

Because of the [[wolf interval]] when using a 12-tone Pythagorean temperament, this tuning is rarely used today, although it is thought to have been widespread. In music which does not change [[key (music)|key]] very often, or which is not very [[harmony|harmonically]] adventurous, the wolf interval is unlikely to be a problem, as not all the possible fifths will be heard in such pieces. In extended Pythagorean tuning there is no wolf interval, all perfect fifths are exactly 3:2.

Because most fifths in 12-tone Pythagorean temperament are in the simple ratio of 3:2, they sound very "smooth" and consonant. The thirds, by contrast, most of which are in the relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth depending on the instrument.<ref>However, 3/2<sup>8</sup> is described as "almost exactly a just major third." Sethares (2005), p.&nbsp;60.</ref>

From about 1510 onward, as thirds came to be treated as consonances, [[meantone temperament]], and particularly [[quarter-comma meantone]], which tunes thirds to the relatively simple ratio of [[sesquiquartum|5:4]], became the most popular system for tuning keyboards. At the same time, syntonic-diatonic [[just intonation]] was posited first by [[Bartolomé Ramos de Pareja|Ramos]] and then by [[Zarlino]] as the normal tuning for singers.

However, meantone presented its own harmonic challenges. Its wolf intervals proved to be even worse than those of the Pythagorean tuning (so much so that it often required 19 keys to the octave as opposed to the 12 in Pythagorean tuning). As a consequence, meantone was not suitable for all music. From around the 18th century, as the desire grew for instruments to change key, and therefore to avoid a wolf interval, this led to the widespread use of [[well temperament]]s and eventually [[equal temperament]].

Pythagorean temperament can still be heard in some parts of modern classical music from singers and from instruments with no fixed tuning such as the [[violin family]]. Where a performer has an unaccompanied passage based on scales, they will tend towards using Pythagorean intonation as that will make the scale sound best in tune, then reverting to other temperaments for other passages (just intonation for chordal or arpeggiated figures, and equal temperament when accompanied with piano or orchestra). Such changes are never explicitly notated and are scarcely noticeable to the audience, just sounding 'in tune'.